From experience with Acer Aspire laptops in the past most if not all of them have suffered from overheating GPU's and this led to them either not starting up again or shutting down randomly. Basically they are designed really bad on the inside when it comes to thermal cooling via heat-sinks etc.
I found from opening them up and with some close inspection that the materials either warp under the heat over a short period of time leaving gaps beteen the surface of the GPU and the heat-sink or the actual GPU solder points come away from the motherboard.
I have tried both the heatgun trick on many and this was really only a temporary fix and i have also tried using copper shims to elevate the gaps between the surface of the GPU and heat-sink. The same failure accord after a while so this leads be to believe that it is just all down to bad design/parts.
2 years may not be a long life-span for a laptop on average but Acer's are just really bad build quality.
On the topic of your battery giving you grief i would try (If you can or haven't already) taking the battery out and running it off the power supply and see if it stays running longer. Apart from that they're just not worth having them repaired or paying for new parts.
Good luck anyway